1993 AHSME Problem 30

Given 0\le x_0<1, let x_n=\left\{ \begin{array}{ll} 2x_{n-1} &\text{ if }2x_{n-1}<1 \\ 2x_{n-1}-1 &\text{ if }2x_{n-1}\ge 1 \end{array}\right. for all integers n>0. For how many x_0 is it true that x_0=x_5?

\text{(A) 0} \quad \text{(B) 1} \quad \text{(C) 5} \quad \text{(D) 31} \quad \text{(E) }\infty

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.